Observation

Proof

For {Ui→X}\{U_i \to X\} any open cover of a paracompact manifold also ∐iUi\coprod_i U_i is paracompact. Hence we may find a differentiably good open cover {Kj→∐iUi}\{K_j \to \coprod_i U_i\}. This is then a refinement of the original open cover of XX.

Definition

Let CartSpsynthdiff↪SmoothAlgop{}_{synthdiff} \hookrightarrow SmoothAlg^{op} be the full subcategory on the objects of the form U×DU \times D with D∈CartSpsmooth↪SmoothAlgopD \in CartSp_{smooth} \hookrightarrow SmoothAlg^{op} and D∈InfPoint↪SmoothAlgopD \in InfPoint \hookrightarrow SmoothAlg^{op}. Write

Proof

This follows as a special case of this proposition after observing that CartSpsynthdiffCartSp_{synthdiff} is an infinitesimal neighbourhood site of CartSpsmoothCartSp_{smooth} in the sense defined there.